FOM: Vaughan Pratt's continua

[Colin McLarty]

Vaughan Pratt writes:

>Here's my own understanding of the tabula rasa that category theorists
>picture themselves starting from when developing category theory
>axiomatically, or at least should if the axioms are to come across as
>common-sensical.

        Vaughan goes on to describe categories *as* structures. For example
the natural numbers become a category    0-->1-->2-->3-->......

        I am interested in foundational uses of categories as structures.
And I think a lot of categorists are, including some in computer science who
think of a data type as a structure which should be described as a category.
I have not thought through Vaughan's analysis in terms of "continua". 

        But I have been defending the perspective of categories *of*
structures: the category of sets, the category of differentiable spaces, et c.

>unlike Colin I believe in starting axiomatically with 
>2-categories at a minimum, i.e. a planar continuum.

        I have no brief against 2-categories. But I don't habitually think
in terms of them.

>I maintain that before one has been corrupted by either camp, both
>notions, collection and continuum, are equally good starting points for
>a foundation of mathematics.  Many (most?) category theorists recognize
>this---as the underdogs today they are in a better position than set
>theorists to do so---and are fond of reflecting wistfully over a beer
>what might have been had only category theory been invented before 1870.

        I got wistful about that once, over coffee with Saunders MacLane,
who called it nonsense. I don't doubt others have done just this with the
beer. Actually, I talked over beer with Vaughan at the ASL meeting in Kansas
City. Did I talk this way then? Did Peter Freyd?

Colin